THE ASTROPHYSICAL JOURNAL, 496 : 605È614, 1998 April 1
( 1998. The American Astronomical Society. All rights reserved. Printed in U.S.A.
BARYONIC FEATURES IN THE MATTER TRANSFER FUNCTION
DANIEL J. EISENSTEIN AND WAYNE HU1
Institute for Advanced Study, Princeton, NJ 08540
Received 1997 August 8 ; accepted 1997 November 6
ABSTRACT
We provide scaling relations and Ðtting formulae for adiabatic cold dark matter cosmologies that
account for all baryon e†ects in the matter transfer function to better than 10% in the large-scale structure regime. They are based upon a physically well-motivated separation of the e†ects of acoustic oscillations, Compton drag, velocity overshoot, baryon infall, adiabatic damping, Silk damping, and cold
dark matter growth suppression. We also Ðnd a simpler, more accurate, and better motivated form for
the zero-baryon transfer function than previous works. These descriptions are employed to quantify the
amplitude and location of baryonic features in linear theory. While baryonic oscillations are prominent if
the baryon fraction ) /) Z ) h2 ] 0.2, the main e†ect in more conventional cosmologies is a sharp
b 0 function
0
suppression in the transfer
below the sound horizon. We provide a simple but accurate description of this e†ect and stress that it is not well approximated by a change in the shape parameter !.
Subject headings : cosmology : theory È dark matter È large-scale structure of universe
1.
INTRODUCTION
and new theoretical interpretations of the Lya forest
(Weinberg et al. 1997, and references therein) suggest a
value of the baryon density ) greater than the Ðducial
nucleosynthesis value of 0.0125b h~2 (Walker et al. 1991).
Meanwhile, observations of large-scale structure (e.g.,
Bahcall 1996), higher Hubble constants (e.g., Freedman
1996), and high-redshift galaxies (e.g., Spinrad et al. 1997)
and clusters (Bahcall, Fan, & Cen 1997) favor a universe
with ) \ 1. Such baryon fractions lead to modiÐcations of
0 CDM transfer function that are within obserthe pure
vational sensitivities (Tegmark 1997 ; Goldberg & Strauss
1997).
Numerical codes to solve the multispecies Boltzmann
equations (e.g., Bond & Efstathiou 1984 ; Holtzmann 1989 ;
Hu et al. 1995 ; Seljak & Zaldarriaga 1996) now agree to 1%
accuracy and run in a few minutes on todayÏs workstations.
While these codes (e.g., the publicly available CMBfast)
should be used for applications demanding high accuracy,
analytic descriptions are useful for understanding how the
di†erent physical e†ects give rise to the behavior seen in the
transfer function. Such descriptions better isolate the
unique and robust observational signatures of physical processes in the early universe and quantify their scalings with
cosmological parameters. They also probe possible parameter degeneracies and suggest possible consistency tests with
related e†ects in the CMB.
To this end, we develop here a Ðtting formula for the
matter transfer function of the general CDM plus baryon
universe (see eqs. [16]È[24] and ° 2). The formula is composed of a number of well-motivated ingredients, whose
behaviors we discuss in detail. We achieve fractional accuracies of 10% in fully baryonic models and D5% in partial
baryon models. Included here is a quantiÐcation of the fundamental scales including the acoustic and Silk damping
scales that are related but not equal to the equivalent scales
in the CMB.
We then use this form to produce quantitative assessments of the amplitude and location of baryonic oscillations, as well as the alteration to the intermediate-scale
shape and small-scale normalization of the transfer function. Previous assessments of the latter e†ects (see, e.g.,
A key success of the cold dark matter (CDM) paradigm is
the ability of linear perturbation theory in the early universe
to explain the power spectra observed in the cosmic microwave background (CMB) and galaxy surveys. On the
largest scales, COBE Ðnds a trend of power with scale
rather close to the theoretically motivated scale-invariant
spectrum (Bennett et al. 1996). On scales between 10 and
200 Mpc, however, galaxy surveys (e.g., Baugh & Efstathiou
1993 ; Strauss & Willick 1995) Ðnd a much di†erent trend, in
which the power increases with scale. Merely by including
the e†ects of the transition between a radiation-dominated
and matter-dominated universe, the CDM cosmology
roughly explains both the relative normalization and the
di†ering spectral indices of these two regimes.
Although the presence of cold dark matter does play a
leading role in determining the matter power spectrum, the
inclusion of baryons can lead to signiÐcant alterations.
Indeed baryonic features in the power spectrum are a fundamental prediction of the gravitational instability paradigm, and their discovery would represent a strong
consistency test for the cosmological model. Such features
are the direct result of small density Ñuctuations in the early
universe prior to recombination. At those times, the
baryons are tightly coupled with the photons and share in
the same pressure-induced oscillations that lead to acoustic
peaks in the CMB. This not only leads to intermediate-scale
oscillations in the power spectrum but also produces an
overall suppression of power on small and intermediate
scales.
While the low baryon fraction (D5%) in the standard
CDM model may have justiÐed the neglect of baryonic
e†ects on the power spectrum in the past, recent observations favor higher baryon fractions. X-ray observations of
clusters of galaxies yield baryon fractions of 10%È30%
(White et al. 1993 ; David, Jones, & Foreman 1995). Recent
measurements of high-redshift deuterium abundances
(Tytler, Fan, & Burles 1996 ; but see Rugers & Hogan 1996)
1 Alfred P. Sloan Fellow.
605
606
EISENSTEIN & HU
TABLE 1
A LIST OF SYMBOLS USED IN THIS PAPER
Symbol
Description
Equation/Section
! ........
!0 . . . . . . .
eff
# ......
a 2.7. . . . . . . .
!
a .........
b
a .........
c
b .........
bb . . . . . . . . .
c
b
......
node
R .........
R ........
T d. . . . . . . . .
T .........
b
T .........
Tc . . . . . . . .
T3 0 . . . . . . . .
0
k..........
k ........
keq . . . . . .
qSilk
..........
s ..........
s8 . . . . . . . . . .
z .........
d
z ........
eq
) h
0
E†ective
shape
Temperature of CMB
! suppression
Baryon suppression
CDM suppression
Baryon envelope shift
Shift in CDM log
Node shift parameter
Photon-baryon ratio
R at z
d function (TF)
Transfer
Baryon sector TF
CDM sector TF
Zero-baryon TF
Pressureless TF
Wavenumber
Equality wavenumber
Silk wavenumber
k scaled with k
Sound horizon eqat drag epoch
E†ective sound horizon
Redshift of drag epoch
Redshift of equality
(28)
(30)
°2
(31)
(14)
(11)
(24)
(12)
(23)
(5)
(6)
(1), (16)
(13), (21)
(9), (17)
(29)
(19)
(1)
(3)
(7)
(10)
(6), (26)
(22)
(4)
(2)
Peacock & Dodds 1994 ; Sugiyama 1995), though sufficient
for the current generation of experiments at low baryon
fraction, are not accurate enough for future high-precision
tests expected of the Sloan Digital Sky Survey (Gunn &
Weinberg 1995) and 2dF Survey (Taylor 1995). We present
an approximate form that neglects the acoustic oscillations
but accurately represents the suppression of power on intermediate scales (eqs. [26] and [28]È[31]). To this end, we
also present a simpler and more accurate formula (eq. [29])
for the zero-baryon transfer function (e.g., Bardeen et al.
1986).
In ° 2 we lay the groundwork for the subsequent discussions by presenting a summary of the physical scales that
enter linear perturbation theory and the exact small-scale
solution (Hu & Sugiyama 1996, hereafter HS96) that we use
to anchor our Ðtting formula. In ° 3 we state the Ðtting
formula and discuss its performance. In ° 4 we describe the
phenomenology revealed by the formula and present simple
scalings to characterize the baryon oscillations and shape
alteration. We conclude in ° 5. In the Appendix, we give a
short guide to help the reader locate relevant formulae from
the paper and turn them into COBE-normalized power
spectra. A list of symbols used in this paper is given in
Table 1.
2.
PHYSICAL EFFECTS
To motivate and explain the transfer function formulae in
° 3, we begin with a review of some of the basic results of
linear perturbation theory, starting with a summary of the
physical scales that enter the theory. We then describe the
exact small-scale solutions found by HS96, as they play a
central role in the development of the Ðtting formulae and
the explanation of baryon phenomenology.
The particular physical properties of the constituents of
the universe, in particular their equations of state and interactions, can alter the predictions of perturbation theory.
Causality, however, precludes e†ects at arbitrarily large
Vol. 496
scales. It is therefore usual to measure the resulting perturbations by comparing them to the amplitude they would
have had were causal physics neglected. The result is the
transfer function, deÐned as2
T (k) 4
d(k, z \ 0) d(0, z \ O)
,
d(k, z \ O) d(0, z \ 0)
(1)
where d(k, z) is the density perturbation for wavenumber k
and redshift z. By construction, T ] 1 as k ] 0. The power
spectrum So d(k) o2T is proportional to the square of the
transfer function multiplied by the initial power spectrum,
most often taken to be proportional to a power law kn with
n B 1. Strictly speaking, each species of particle has a
separate transfer function ; however, after recombination
the baryons are essentially pressureless and quickly catch
up with the cold dark matter perturbations, leaving both
with the same transfer function. It is this transfer function
that we study in this paper.
We consider cosmologies in which the universe is primarily composed of photons, baryons (and their accompanying electrons), massless neutrinos, and cold dark
matter (CDM). Relative to the critical density, the densities
today of the baryons and CDM are ) and ) , respectively.
b ) ])
c . The CMB
The total matter density is then ) \
0
b
temperature T
is written as 2.7# K ; thec best obserCMB
2.7 instrument are
vations from the COBE FIRAS
2.728 ^ 0.004 K (Fixsen et al. 1996 ; 95% conÐdence level).
We assume that the massless neutrinos contribute an
energy density corresponding to three species at (4/11)1@3
the temperature of the photons. We use a Hubble constant
H and deÐne h 4 H /(100 km s~1 Mpc~1). It is important
to0note that for z ? 0)~1 the dynamics of the Ñuctuations
0 by the matter-radiation ratio
are determined solely
) h2#~4, the baryon-photon ratio ) h2#~4, and the
0
2.7
2.7 to be the
CMB
temperature
T . We Ðx the last bof these
CMB
COBE value and do not include variations in # in our
Ðts. Since all e†ects in the transfer function are set2.7at those
early times, the resulting description should depend only on
) h2 and ) /) . The existence today of a nonzero cosmo0
b 0or curvature is insigniÐcant.
logical
constant
2.1. L ength and T imescales
The physics governing the evolution of perturbations in
CDM-baryon universes involves three distinct length
scales : the horizon size at matter-radiation equality, the
sound horizon at the time of recombination, and the Silk
damping length at recombination.
In the usual cosmological paradigm, nonrelativistic particles (baryons, electrons, and CDM) dominate relativistic
particles (photons and massless neutrinos) in density today.
However, because the density of these two classes of particles scale di†erently in time, at an earlier time, the reverse
situation held. The transition from a radiation-dominated
universe to a matter-dominated one occurs roughly at
z \ 2.50 ] 104) h2#~4 ,
(2)
eq
0
2.7
the redshift where the two classes have equal density. As
density perturbations behave di†erently in a radiation-
2 In the synchronous or comoving gauge ; gauge issues of are no practical concern for subhorizon scales.
No. 2, 1998
BARYONIC FEATURES IN THE MATTER TRANSFER FUNCTION
dominated universe versus a matter-dominated one due to
pressure support, the scale of the particle horizon at the
equality epoch z ,
eq
k 4 (2) H2 z )1@2 \ 7.46 ] 10~2) h2#~2 Mpc~1 ,
eq
0 0 eq
0
2.7
(3)
is imprinted on the matter transfer function ; in particular,
perturbations on smaller scales are suppressed in amplitude
in comparison to those on large scales. If the universe consisted only of noninteracting matter and radiation, the
matter transfer function would depend on the ratio (k/k )
eq
alone.
Complications arise due to interactions between the
species. Prior to the recombination of baryons and electrons, the large density of free electrons couples the baryons
to the photons through Coulomb and Compton interactions so that the three species move together as a single
Ñuid. This continues until, in the process of recombination,
the rate of Compton scattering between photons and electrons becomes too low, freeing the baryons from the
photons. We thus deÐne the drag epoch z as the time at
d
which the baryons are released from the Compton
drag of
the photons in terms of a weighted integral over the
Thomson scattering rate (see HS96, eqs. [C8], [E2]). A Ðt to
the numerical recombination results is
() h2)0.251
0
[1 ] b () h2)b2] ,
z \ 1291
1 b
d
1 ] 0.659() h2)0.828
0
b \ 0.313() h2)~0.419[1 ] 0.607() h2)0.674] ,
1
0
0
b \ 0.238() h2)0.223 ,
(4)
2
0
where we have reduced z by a factor of 0.96 from HS96 on
d
phenomenological grounds.
For ) h2 [ 0.03, this epoch
follows last scattering of the photons.b
Prior to z , small-scale perturbations in the photond
baryon Ñuid propagate
as acoustic waves. The sound speed
is c \ 1/[3(1 ] R)]1@2 (in units where the speed of light is
s where R is the ratio of the baryon to photon momenunity),
607
tum density,
R 4 3o /4o \ 31.5) h2#~4(z/103)~1 .
(5)
b c
b
2.7
We deÐne the sound horizon at the drag epoch as the comoving distance a wave can travel prior to redshift z
d
t(zd)
s\
c (1 ] z)dt
s
0
6
J1 ] R ] JR ] R
2
(6)
d
d
eq ,
ln
\
R
3k
1
]
JR
eq
eq
eq
where R 4 R(z ) and R 4 R(z ) are the values of R at the
d
d
eq
eq
drag epoch and epoch of matter-radiation equality, respectively. The sound horizon at the drag epoch (hereafter
simply the sound horizon) is larger than the equality
horizon (D1/k ) in high-) models but smaller than it is in
eq
0
low-) models ; it also decreases strongly with increasing
0
baryon fraction if R Z 1 (see Fig. 1).
On small scales, dthe coupling between the baryons and
the photons is not perfect, such that the two species are able
to di†use past one another (Silk 1968). The Silk damping
scale is well Ðtted by the approximation
P
S
k \ 1.6() h2)0.52() h2)0.73[1 ] (10.4) h2)~0.95]
Silk
b
0
0
Mpc~1 ,
(7)
which represents a ^20% phenomenological correction
from the value given in HS96. The Silk scale is generally a
smaller length scale than either s or 1/k . Note that the
eq
di†erence between the drag and last scattering
epochs
implies that for ) h2 [ 0.03 the sound and Silk scales in the
transfer function bare larger than those in the CMB. We
show a comparison of these scales as a function of cosmological parameters in Figure 1.
2.2. Small-Scale Solutions
In the small-scale limit, one can solve the linear perturbation equations analytically in the approximation that
baryons provide no gravitational source to the CDM
(HS96). This approximation is appropriate below the sound
FIG. 1.ÈComparison of the physical scales as functions of ) h2 and the baryon fraction ) /) . (a) The equality scale vs. the sound horizon : k s/n
0 vs. the Silk scale : k s/n (unlabeled
b 0
(unlabeled contours are at 0.1 increments). (b) The sound horizon
contours are 2 and 3). The factors of n have eqbeen
Silk
included to facilitate comparison to the acoustic scale.
608
EISENSTEIN & HU
horizon since baryon perturbations are pressure supported.
As we will use this solution in order to anchor the smallscale end of our Ðtting formulae, we describe the solutions
further.
The transfer function is written as a sum of the baryon
and cold dark matter contributions at the drag epoch
)
)
T (k) \ b T (k) ] c T (k) .
(8)
b
)
) c
0
0
The CDM transfer function can be solved exactly in terms
of hypergeometric functions that are more conveniently
approximated by the following form :
q\
A
ln 1.8b q
c ,
T ]a
c
c 14.2q2
(9)
B
k
k
#2 () h2)~1 \
Mpc~1 2.7 0
13.41k
,
eq
(10)
where a and b are Ðtted by
c
c
a \ a~)b@)0a~()b@)0)3 ,
2
c
1
a \ (46.9) h2)0.670[1 ] (32.1) h2)~0.532] ,
1
0
0
a \ (12.0) h2)0.424[1 ] (45.0) h2)~0.582] , (11)
2
0
0
b~1 \ 1 ] b [() /) )b2 [ 1] ,
c
1 c 0
b \ 0.944[1 ] (458) h2)~0.708]~1 ,
1
0
b \ (0.395) h2)~0.0266 .
(12)
2
0
As ) /) ] 0, a , b ] 1. Equation (9) shows the familiar ln
0
c ofc the small-scale CDM transfer function.
(k)/k2b dependence
This occurs because outside the horizon, density perturbations grow as k2 due to the product of potential and
velocity gradients that drive the growth ; inside the horizon
in the radiation-dominated epoch the growth is logarithmic.
The main e†ect of the baryons comes from the suppression
in growth rates between equality and the drag epoch. As
) h2 increases, the time between the two epochs increases ;
0 the maximum suppression due to the baryons occurs in
thus
the highest ) h2 models. A plot of a ) /) is shown in
0
c c 0
Figure 2a.
In the small-scale limit, the baryons are trapped in acoustic oscillations until recombination permits them to slip
past the photons. While the density perturbation of this
oscillation contributes to the transfer function, the corre-
Vol. 496
sponding velocity perturbation actually dominates in the
small-scale limit. When the oscillation is released at the
drag epoch, the baryons move kinematically according to
their velocity and generate a new density perturbation
(Sunyaev & Zeldovich 1970 ; Press & Vishniac 1980). This
so-called velocity overshoot means that the transfer function for ks ? 1 follows
T ]a
b
b
sin(ks)
D(k) .
ks
(13)
Here D(k) represents the e†ects of Silk damping, which
occurs due to combination of di†usion of the photons with
respect to the baryons and Compton drag moving baryons
from overdensities to underdensities and hence destroying
the perturbation. That the dependence is sin(ks) rather than
cos(ks) is the result of the dominance of the velocity term
rather than the density term. A detailed treatment allows
one to calculate a :
b
1]z
eq ,
a \ 2.07k s(1 ] R )~3@4G
(14)
b
eq
d
1]z
d
J1 ] y ] 1
. (15)
G(y) \ y [6J1 ] y ] (2 ] 3y) ln
J1 ] y [ 1
A
C
A
B
BD
The factor (1 ] R )~3@4 comes from the damping of oscild
lations resulting from
the adiabatic decrease in the sound
speed (Peebles & Yu 1970 ; HS96, eq. [A17]). The factor
involving G(y) (Py~1@2 for y ? 1) comes from the product
of the growth suppression between equality and the drag
epoch (Py~1) and the time available before the velocities
creating the perturbation decay due to the free expansion of
the universe (Py1@2). A plot of a ) /) is shown in Figure
b b 0
2b.
That the phase of the oscillations is ks (Peebles & Yu
1970) can be seen simply from integrating the dispersion
relation u \ kc . The change in phase for an acoustic wave
is d/ \ [k(1 ] sz)]c dt. Integrating this from t B 0 to the
drag epoch (where sthe baryons are released and the oscillations freeze out) yields ks, owing to the deÐnition in
equation (6). A technical complication occurs for k Z k .
Silk
The presence of strong damping slightly raises the redshift
at which the oscillations freeze out, making s a few percent
smaller (see HS96 Fig. 2). We have neglected this e†ect
because it occurs at sufficiently small scales that the
FIG. 2.ÈSuppression factors for (a) the CDM (a ) /) ) and (b) the baryonic acoustic oscillations (a ) /) )
c c 0
b b 0
No. 2, 1998
BARYONIC FEATURES IN THE MATTER TRANSFER FUNCTION
609
FIG. 3.ÈFour examples of the Ðt compared to numerical results. The larger plots show the numerical result (solid lines) and the Ðt (dashed lines). The
smaller subplots show the residuals, deÐned as the di†erence between the two divided by a nonoscillatory envelope. Note that in the fully baryonic models,
the oscillations have alternating sign in the transfer function. Also shown is the zero-baryon case (dotted lines) ; note the strong suppression on scales below
the sound horizon due to the baryons.
resulting phase shift is unobservable in practice, but one can
see the deviations when comparing to numerical results (see
Fig. 3).
3.
FITTING FORMULAE
As we have seen in ° 2, analytic solutions exist for the
transfer function at both large and small scales. The transition between these extremes is deÐned by two scales, the
horizon at matter-radiation equality and the sound horizon
at the end of the drag epoch. The former sets the dynamics
of the expansion and perturbation growth ; the latter sets
the scale at which pressure support becomes important for
the baryons. Because the range of scales accessible by the
study of structure formation falls within this transition
regime, it is important to understand the full transfer function in detail. To that end, we present in this section a Ðtting
formula that approximates the full transfer function on all
scales.
We write the transfer function as the sum of two pieces,
)
)
T (k) \ b T (k) ] c T (k) ,
(16)
) b
) c
0
0
whose origins lie in the evolution before the drag epoch of
the baryons and cold dark matter, respectively. This separation is physically reasonable, as before the drag epoch the
two species were dynamically independent and after the
drag epoch their Ñuctuations are weighted by the fractional
density they contribute. This automatically includes in T
the e†ects of baryonic infall into CDM potential wells. Notec
however that T and T are themselves not true transfer
b do notc reÑect the density perturbations of
functions, as they
the relevant species today. Rather, it is their densityweighted average T (k) that is the transfer function for both
the baryons and the CDM.
3.1. Cold Dark Matter
The transfer function for cosmologies in which noninteracting cold dark matter dominates over baryons has been
studied by many authors, and accurate Ðtting formulae
already exist in this limit (e.g., Bond & Efstathiou 1984 ;
Bardeen et al. 1986 ; but see improvements in our ° 4.2).
However the e†ect of baryons, though long known from
numerical calculations (e.g., Peebles & Yu 1970 ; Holtzmann
1989), have in the past either been included in Ðtting formulae in an ad hoc manner (see, e.g., Peacock & Dodds 1994 ;
Sugiyama 1995) or only in the small-scale limit (HS96).
In the presence of baryons, the growth of CDM perturbations is suppressed on scales below the sound horizon.
The change to the asymptotic form can be calculated and
has been shown in equations (9)È(12). We introduce this
suppression by interpolating between two solutions near
the sound horizon :
T (k) \ fT3 (k, 1, b ) ] (1 [ f )T3 (k, a , b )
c
0
c
0
c c
(17)
610
EISENSTEIN & HU
f\
1
,
1 ] (ks/5.4)4
(18)
with
ln (e ] 1.8b q)
c
T3 (k, a , b ) \
,
(19)
0
c c
ln (e ] 1.8b q) ] Cq2
c
14.2
386
C\
]
.
(20)
a
1 ] 69.9q1.08
c
The variables q, a , and b have been given in equations (10),
c
c
(11), and (12), respectively.
3.2. Baryons
In the case of cosmologies without cold dark matter, the
transfer function departs from unity below the sound
horizon to exhibit a series of declining peaks due to acoustic
oscillations. The small-scale exact solution of equation (13)
suggests that these may be written as the product of a
declining oscillatory term, a suppression due to the decay of
potentials between the equality and drag scales, and an
exponential Silk damping. We therefore write
C
D
T3 (k ; 1, 1)
a
0
b
]
e~(k@kSilk)1.4 j (ks8 ) .
T \
0
b
1 ] (ks/5.2)2 1 ] (b /ks)3
b
(21)
Here the spherical Bessel function j (x) 4 (sin x)/x is a piece
0
that approaches unity above the sound
horizon but oscillates below it. The envelope in square brackets traces the
zero-baryon CDM case above the sound horizon and then
breaks to a constant multiplied by an exponential Silk
damping factor. We attach the Silk damping factor only to
the second term because such di†usion can only occur on
scales below the sound horizon, where only the second term
contributes signiÐcantly. This subtlety marginally improves
the Ðt. The sound horizon s, Silk scale k , and amplitude
suppression a were given in equationsSilk
(6), (7), and (14),
b now discuss s8 and b .
respectively ; we
b
While the nodes of the baryonic transfer
function asymptotically approach those of sin(ks) for ks ? 1, the Ðrst few
nodes (ks [ 10) fall at higher k than predicted by sin(ks).
This shift is due to the contribution of the baryon density
perturbation itself at the drag epoch and reÑects the fact
that at the sound horizon velocity overshoot is not the
dominant e†ect. This e†ect increases with ) h2 because the
time available for velocity overshoot (see eq.0[14]) decreases
as (z /z )1@2 and is only weakly dependent on the baryon
d eqWe have veriÐed this explanation of the node shift
fraction.
by isolating the density and velocity contributions of the
baryons at the drag epoch from numerical evolution codes.
We address this shifting of the nodes phenomenologically
by introducing the quantity
s8 (k) \
s
.
(22)
/ks)3]1@3
node
For ks ? b , s8 ] s, restoring the sinusoidal nodes.
However, atnode
ks [ b , s8 B ks2/b
\ s, moving the nodes
node
to higher k. We Ðndnode
[1 ] (b
b
\ 8.41() h2)0.435 ,
(23)
node
0
independent of the baryon fraction. Hence the e†ect gets
smaller at low ) as expected.
0
Vol. 496
The amplitude a speciÐes the small-scale asymptotic
b
contribution of the velocity portion of the acoustic oscillation. Two e†ects modify this amplitude at large scales.
Above the sound horizon, velocity contributions to the
transfer function fall o†. Furthermore, the amplitude
declines if CDM dominates the energy density of the
photon-baryon system when the wavelength enters the
horizon. This occurs due to the absence of feedback in the
gravitational driving of the photon-baryon oscillator
(HS96, ° 3.1). The net result is a cuto† associated with the
sound horizon that moves to smaller scales as ) h2
increases and/or ) /) decreases. We describe this0 in
b 0
equation (21) by turning on the velocity term at the characteristic scale b s, where
b
)
)
b \ 0.5 ] b ] 3 [ 2 b J(17.2) h2)2 ] 1 . (24)
b
0
)
)
0
0
A
B
3.3. Performance
For the parameter range 0.025 [ ) h2 [ 0.25 and 0 ¹
0
) /) ¹ 1 the Ðtting formula works quite
well. For fully
b 0
baryonic
models (i.e., ) \ 0) the fractional residuals are
nearly always under 10%.c As the baryon fraction decreases,
the accuracy improves due to the increasing contribution of
the simpler CDM piece. Residuals smaller than 5% are
typical for ) /) \ 0.5. Note that we quote the residuals as
b between
0
the di†erence
the Ðt and the numerical results
divided by a nonoscillatory envelope that is deÐned by
replacing (sin ks8 )/ks8 in equation (21) by [1 ] (ks8 )4]~1@4.
This envelope matches the knee of the transfer function and
grazes all the subsequent maxima.
In Figure 3 we display four comparisons of the Ðtting
formula relative to the numerical results. Also shown are
the residuals relative to our envelope. The reason for the
degradation in the Ðt at the shortest scales is that small
errors in the sound horizon (see the end of ° 2.2) produce
signiÐcant errors in the phase of the oscillations, producing
order unity residuals. However, the Ðtting formula reproduces the correct amplitude and hence the Silk scale.
The most serious systematic error in the Ðtting formula
occurs for ) h2 Z 0.25. In the baryon sector of these cases,
0
the drop between
ks B 1 and the oscillations at ks Z 5
becomes quite precipitous. Our formula does not decline
this quickly, causing the amplitude of the Ðrst valley to be
signiÐcantly overestimated. Later peaks and valleys are
underestimated in an attempt to compensate. One can see
the beginnings of this trend in the ) \ ) \ 1 example in
0 for higher
b
Figure 3 ; the problem gets more severe
) h2.
A less important systematic e†ect occurs for high0 baryon
fraction (0.7 \ ) /) \ 0.9), low-) models. Because of a
0 break scale (eq. [18]) that we have
small shift in the bCDM
chosen not to model, the Ðrst valley is systematically underestimated by D10%È15% in amplitude. This problem does
not extend to lower baryon fractions.
4.
PHENOMENOLOGY
There are a number of phenomenological trends as a
function of cosmological parameters. The two basic e†ects
that arise from the inclusion of baryons are the introduction
of oscillations and the suppression of power below the
sound horizon, with a corresponding sharpening of the
bend around the sound horizon. We discuss these two in
turn.
No. 2, 1998
BARYONIC FEATURES IN THE MATTER TRANSFER FUNCTION
611
where
44.5 ln (9.83/) h2)
0
Mpc
(26)
J1 ] 10() h2)3@4
b
approximates the sound horizon to D2% over the range
) h2 Z 0.0125 and 0.025 [ ) h2 [ 0.5. The value of k
as
b
0
peak
a function of cosmological parameters is shown in Figure 4.
The amplitude of the oscillations also has a nontrivial
dependence on the cosmological parameters. The oscillations of course grow stronger as the baryon fraction
increases. However, at Ðxed baryon fraction, they are
weaker compared to CDM contributions in high ) h2 due
to the increase in the time available for the CDM 0to grow
between equality and the drag epoch. While the full series of
peaks and valleys may be impossible to observe due to
nonlinear structure formation, the Ðrst valley and peak are
generally in the linear regime. In Figure 5 we show the
fractional enhancement of power due to the oscillations
over the smooth CDM contributions. The Ðrst peak grows
monotonically with the baryon fraction. The Ðrst trough is
more subtle : while the transfer function at this location
simply declines as the baryon fraction increases, when T (k)
goes negative, the power, which is the square of T , will
regenerate. Perfect cancellation of the baryon and CDM
contributions occurs along the contour labeled ““ [1 ÏÏ in
Figure 5a ; above this line the trough in amplitude becomes
a peak in power as the baryon contributions come to fully
dominate. A useful rough scaling as to when oscillations
become important is given by
s\
FIG. 4.ÈLocation of the Ðrst peak in Mpc~1 as a function of cosmological parameters.
4.1. Baryon Oscillations
Two interesting aspects of the baryonic oscillations are
the location and amplitude of the peaks and troughs. Well
under the sound horizon we expect them to fall at
k \ mn/2s where m \ 3, 7, 11, . . . for troughs and m \ 5, 9,
13, . . . for peaks. However, as described in ° 3.2, the Ðrst few
oscillations are shifted according to the parameter b . As
node
) h2 increases, the Ðrst few peaks and troughs are progres0
sively shifted to higher k. A corollary to this shift is that the
ratio of the node locations becomes smaller as one raises
) , i.e., the valleys and peaks become slightly narrower. The
0
location
of the Ðrst peak is conveniently Ðtted as
k
peak
\
5n
(1 ] 0.217) h2) ,
0
2s
(25)
)
b Z ) h2 ] 0.2 ,
(27)
0
)
0
which crudely describes the region where the change in
power is greater than D20%.
A related trend is the increase with ) h2 of the sharpness
0 the knee at ks B 1
of the decline in the baryonic sector from
(T B 1) into the series of oscillations below the sound
horizon (Peebles 1981). This is most easily seen in the fully
baryonic models of Figure 3 ; near k \ 0.1 h Mpc~1 in the
) h2 \ 0.25 model, the transfer function drops a factor of
100 in under half a decade in k. This break becomes even
more striking in higher ) h2 cases.
0
FIG. 5.ÈFractional enhancement of power due to (a) the Ðrst valley and (b) the Ðrst peak relative to nonoscillatory CDM portion of the transfer function
(T /T )2 [ 1, at the appropriate wavevector from the Ðt. Unlabeled contours are at (a) [0.8, 0, 1, 3 and (b) 8.0, 16.0.
c
612
EISENSTEIN & HU
Vol. 496
4.2. E†ective Shape
As we have seen, if ) /) [ ) h2 ] 0.2, the main e†ect
b 0
0
of the baryons is not to introduce oscillations into the transfer function but to suppress the k~2 tail from the growth of
CDM perturbations. This occurs both because the CDM
portion T is suppressed by a and because the baryonic
c
portion Tc is providing essentially
no power below the
b
sound horizon. As noted just above, the latter transition can
occur quite quickly. These suppressions indicate that the
shape of the transfer function must change, with a break
near the sound horizon. It is useful to quantify this e†ect.
Let us work forward from the zero-baryon case. Here the
transfer function is parameterized by q P k/k , more comeq
monly expressed as a shape parameter ! \ ) h, where
0
k
q\
#2 /! .
(28)
h Mpc~1 2.7
A commonly used Ðtting formula to the zero-baryon limit
was presented by Bardeen et al. (1986, eq. [G3]). However
this formula Ðts neither the exact small-scale solution of
° 2.2 nor does it have the quadratic deviation from unity
required by the theory. The latter is a fundamental requirement of causality (Zeldovich 1965), in that one power of k
must arise from stress gradients generating bulk velocity
and a second from velocity gradients generating density
perturbations. In fact the coefficient of this quadratic deviation can be calculated perturbatively if the stress gradients
are dominated by the isotropic (pressure) term.
The following functional form satisÐes these criteria and
is a better Ðt to the zero-baryon case extrapolated from
trace-baryon models calculated by CMBfast
L
0
,
T (q) \
0
L ] C q2
0
0
L (q) \ ln (2e ] 1.8q) ,
0
731
.
C (q) \ 14.2 ]
0
1 ] 62.5q
FIG. 6.ÈComparison of various zero-baryon transfer functions to the
numerical calculation of CMBfast (Seljak & Zaldarriaga 1996). The Ðtting
form of eq. (29) agrees with CMBfast to 1% ; whereas previous approximations Peebles (1982 ; P82), Starobinsky & Sahni (in Modern Theoretical
and Experimental Problems of General Relativity, as quoted in Shandarin
& Zeldovich 1984 ; SS84), and Bardeen et al. (1986 ; BBKS86). Also shown
is the form of Bond & Efstathiou (1984 ; BE84), a 3% baryon fraction
model that has often been used as a zero-baryon proxy.
neglects the oscillations caused by the baryons, the transfer
function is suppressed and roughly follows that of a rescaled
!. Hence the transition around the sound horizon cannot
be Ðtted by a single !.
A reasonable Ðt to the nonoscillatory part of the transfer
function can be written by rescaling ! (k) as one moves
eff
through the sound horizon
C
D
1[a
!
,
! (k) \ ) h a ]
! 1 ] (0.43ks)4
eff
0
(29)
Note that this form is not only more accurate than the
Bardeen et al. (1986) one but is also simpler : there are fewer
parameters and the coefficients 1.8 and 14.2 are derived
theoretically. The parameter 731, the small-q quadratic
deviation, has been Ðtted rather than derived to account for
the small correction due to anisotropic stress gradients. In
Figure 6 we show a comparison of this form to numerical
calculations and various Ðtting formulae in the literature.
Our formula agrees with numerical calculations at the same
level as di†erent numerical calculations agree with each
other (Hu et al. 1995), i.e., to 1% through the CMB and
large-scale structure regimes.
The presence of baryons has commonly been included by
Ðtting a constant shape parameter ! (Peacock & Dodds
1994 ; Sugiyama 1995). That such an approach can work on
small scales can be seen as follows. On small scales, the
e†ect of the baryons is a constant suppression by the factor
a ) /) . Since the transfer function there is proportional to
c c )~2,
0 a rescaling of ! approximates this e†ect.
(k/!
0
However, this simple rescaling0of ! does not properly treat
0
the region observable through large-scale
structure. Above
the sound horizon, the baryons and CDM are indistinguishable, and so the transfer function is close to that
parameterized by ! 4 ) h. Below the horizon, if one
0
0
(30)
)
a \ 1 [ 0.328 ln (431) h2) b
!
0
)
0
) 2
(31)
] 0.38 ln (22.3) h2) b .
0
)
0
DeÐning q as in equation (28), we Ðnd that T (k) B T (q ).
eff
0 seffin
Figure 7 shows
an example. Note the simpler form of
equation (26) may be used here. The quantity a is nearly
(a ) /) )1@2, the radicand of which is plotted in!Figure 2 ;
c 0 the above form for simplicity and to account for
wec provide
small deviations at the higher ) h2 values. The latter arise
0 in the observable region
because the Ðt has been optimized
k [ 0.1 Mpc~1, where the CDM transfer function is not
quite in its k~2 small-scale limit. Of course, neglecting the
oscillatory contributions is not a good approximation for
) /) Z 0.5. In addition to the obvious omission of the
b 0 we have Ðtted a to the transfer function T (k) and
wiggles,
! This neglects the power intronot the observable T (k)2.
duced by the square of the oscillatory term.
We caution the reader that the small-scale asymptote
! \ a ) h arises from the normalization rather than
eff and
! therefore
0
shape
should not be conÑated with ! derived
from redshift surveys (Peacock & Dodds 1994). Pure normalization distinctions are not observable with current redshift surveys ; instead, one estimates ! by Ðtting power
spectra of arbitrary normalization and relying on the di†erences in shape and power-law slope introduced by the
A B
No. 2, 1998
BARYONIC FEATURES IN THE MATTER TRANSFER FUNCTION
FIG. 7.ÈDistortions to the shape of the transfer function as measured
by the true transfer function over the zero-baryon form. The shape function of eq. (30) adequately describes the true curve apart from the oscillations ; whereas the constant shift in ! described by Peacock & Dodds
(1994 ; PD94) and Sugiyama (1995 ; S95) do not.
variation in !. However, when combined with the COBE
normalization (e.g., Bunn & White 1997), the normalized
predictions from equations (30) and (31) are appropriate.
However, equation (30) and Figure 7 do make clear that a
simple rescaling of ! is not accurate near the sound horizon.
Future measurements of large-scale structure expected from
the Sloan Digital Sky Survey (Gunn & Weinberg 1995) and
2dF Survey (Taylor 1995) will be sufficiently precise as to
detect the deviations from a single ! model (Tegmark 1997 ;
Goldberg & Strauss 1997). A key signature is that the bend
from T B 1 to T P k~2 becomes sharper as baryons are
introduced. Detecting this change will require that scales
around s/n (see eq. [26]) are well observed.
5.
DISCUSSION
We have presented an accurate, well-motivated Ðtting
form for the transfer function of a general CDM-baryon
universe. The formula is generally accurate to better than
10% in fully baryonic universes and better than 5% in cosmologies with ) /) [ 0.5. While the available numerical
b yet
0 more accurate transfer functions, the
codes will provide
Ðtting form here should be useful for characterizing trends
613
in cosmological parameter space. Moreover, by separating
the various physical aspects in the analytic form, we hope to
provide physical intuition for the e†ects, their interrelationships, and their correspondence with analogous e†ects
in the CMB.
As applications of the form, we gave quantitative assessments of the location and amplitude of the baryonic features in the linear regime. We quantiÐed the suppression of
power on scales below the sound horizon due to the admixture of baryons and showed that this suppression is not well
Ðtted by a rescaling of ! if one is probing scales near the
sound horizon. An alternative model, based on an interpolation in !, gives a reasonable Ðt to the nonoscillatory
portion of the transfer function provided that ) /) [
b zero0
) h2 ] 0.2. Finally, we gave a new Ðtting form for the
0
baryon limit that is more accurate on small scales than
those used previously. A summary of how to use these
various formulae is given in the Appendix.
Baryonic features, like the acoustic peaks in the CMB
(Hu & White 1996), transcend the adiabatic CDM paradigm discussed in this paper. In fact, any gravitational
instability model where reionization took place no earlier
than z D 270() h2)1@5, the Compton drag epoch for a fully
ionizedd universe,0 must possess acoustic e†ects at some level.
Evidence of these e†ects is strong indication that Ñuctuations were generated in the early universe. Moreover, when
combined with CMB observations, baryonic features allow
strong consistency tests for the predicted growth of Ñuctuations as both sets of features reÑect the underlying acoustic
oscillations before recombination. A measurement of the
sound horizon from the matter power spectrum combined
with its angular extent from the CMB would allow an
angular diameter distance test for curvature in the universe
that is largely free of cosmological assumptions. Baryonic
features in the matter power spectrum thus represent a valuable resource for cosmological information, but one that
may be difficult to mine observationally.
We thank S. Boughn, U. Seljak, J. Silk, D. Spergel, A.
Szalay, M. Tegmark, and M. White for useful discussions.
The CMBfast package3 was used to generate numerical
transfer functions. W. H. acknowledges support from the
W. M. Keck foundation and the hospitality of the Aspen
Center for Physics. D. J. E. and W. H. acknowledge support
from NSF PHY-9513835.
3 http ://arcturus.mit.edu :80/Dmatiasz/CMBFAST/cmbfast.html.
APPENDIX
A USERÏS GUIDE
In this paper, we have presented a Ðtting form for the transfer function in a CDM and baryon cosmology with adiabatic
perturbations. The formula is given as equations (2)È(7), (10)È(12), and (14)È(24). Alternatively, if one prefers a simpler form
that accurately represents the baryon-induced suppression on intermediate scales but that ignores the acoustic oscillations in
the transfer function, one may use the form given in equations (26) and (28)È(31). The oscillations are fairly small ([20%
modulation in power) for ) /) [ ) h2 ] 0.2, so in these cases the simpler form will be appropriate for many applications.
0 accurate form for the zero-baryon transfer function.4
We also provide in equationb(29)0 a more
The power spectrum of the density Ñuctuations is then proportional to the initial power spectrum times the square of the
transfer function. In the most usual case, the initial power spectrum is taken to be a power law, so that P(k) P knT 2(k), where
n \ 1 is the familiar Harrison-Zeldovich-Peebles scale-invariant case.
4 Electronic versions of the formulae in this paper may be found at http ://www.sns.ias.edu/Dwhu/transfer/transfer.html.
614
EISENSTEIN & HU
While the transfer function is independent of late-time e†ects such as the presence of a cosmological constant or curvature,
the magnitude and time dependence of the normalization of the power spectrum does depend on these e†ects. Bunn & White
(1997) calculate the present-day normalization of the power spectrum implied by the 4 year COBE anisotropy measurement
and provide the following Ðtting forms for Ñat and open cosmologies :
A B
ck 3`n
k3
T 2(k) ,
(A1)
*2(k) o
4
P(k) \ d2
z/0 2n2
H H
0
d \ 1.95 ] 10~5)~0.35~0.19 ln )0~0.17“e~“~0.14“2 (" \ 0) ,
(A2)
H
0
2
(A3)
d \ 1.94 ] 10~5)~0.785~0.05 ln )0e~0.95“~0.169“ (" \ 1 [ ) ) ,
0
H
0
where n8 \ n [ 1 and the contribution of tensor perturbations to the observed anisotropies has been assumed to be zero (see
Bunn & White 1997 for more details). The 1 p statistical uncertainty is 7%, and the error in the above Ðts are much smaller
than this for 0.2 ¹ ) ¹ 1 and 0.7 ¹ n ¹ 1.2.
0
To extend this normalization
to earlier times, one needs to scale the power spectrum by the square of the growth function
D (z) (Peebles 1980). As is well known, for ) \ 1, D \ (1 ] z)~1 at redshift z. For other cosmologies, one can use the
1
0
1
approximation (Lahav et al. 1991 ; Carroll, Press, & Turner 1992)
D (z) \ (1 ] z)~1
1
G
C
)(z)
5)(z)
)(z)4@7 [ ) (z) ] 1 ]
"
2
2
DC
) (z)
1] "
70
DH
~1
.
(A4)
Here )(z) and ) (z) are the density parameters as seen by an observer at redshift z ; hence
"
) (1 ] z)3
0
)(z) \
,
(A5)
) ] ) (1 ] z)2 ] ) (1 ] z)3
"
R
0
)
"
) (z) \
,
(A6)
"
) ] ) (1 ] z)2 ] ) (1 ] z)3
"
R
0
where ) is the matter density in units of critical density, ) \ "/3H2 represents the cosmological constant ", and
"
0
) \ 1 [0 ) [ ) represents the e†ects of curvature.
RThe normalization
0
" of the power spectrum is such that the variance of mass Ñuctuations inside a sphere of radius R is
p2(R) \
P
C
D
= dk
3j (kR) 2
*2(k) 1
,
k
kR
0
(A7)
where j (x) \ (sin x [ x cos x)/x2.
1
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